Two Fair Coin Flips Suffice to Beat Distortion Three in Metric Voting

We show that two fair coin flips suffice to improve on the optimal deterministic distortion of three in ordinal metric voting. Given any complete ranking profile, one can fix four equiprobable candidate slots, allowing repeats, whose expected social cost is at most 2.97029814 times that of any original candidate under every compatible metric. We prove existence by coupling an activated source lottery with a nonlinear envelope for the empirical distribution of four draws. An exact signed expectation bound makes the same list valid across metrics. Rational certificates verify the threshold inequalities on their full continuous domain and the four-draw expectation.